In this paper, we propose a new approach towards the classification of spherical fusion categories by their Frobenius–Schur exponents. We classify exponent 2 up to monoidal equivalence. also modular braided It turns out that Gauss sum is complete invariant for 2. This result can be viewed as categorical analog Arf's theorem on non-degenerate quadratic forms over fields characteristic

Let $\mathcal{C}$ be a $(d+2)$-angulated category. In this note, we show that if is locally finite, then has Auslander-Reiten $(d+2)$-angles. This extends result of Xiao and Zhu for triangulated categories.

When I offered to talk here, I asked myself if I had something suitable for a nice short 20 minute talk, and sent in an abstract about “cores of spaces and spectra”. However, since I have been given 50 minutes, I have decided to scrap that for something a little more substantial. It is brand new. I assume that I am among friends and that you won’t mind if some of what I say turns out to be nons...

A general framework for cluster tilting is set up by showing that any quotient of a triangulated category modulo a tilting subcategory (that is, a maximal oneorthogonal subcategory) carries an induced abelian structure. These abelian quotients turn out to be module categories of Gorenstein algebras of dimension at most one.